How to extrude X and Y axis on a traced object (flatten)
Not sure I understand you correctly, I am assuming you want a wider on top, narrower on the bottom section for the cookie cutter, one cutting side sharper edge, and one wider side to make pressure, is that correct?

If so, I'd say it would probably be easier to design the section you desire as a separate curve object, then use it as bevel object in your chicken curve.



It would probably work best if your chicken was a "single stroke curve" instead of a double curve with a designed stroke width.

Otherwise if that is not possible (I see bifurcating strokes in there) you may achieve a similar effect using a "half section" instead. Just make sure the wider part of the section is always narrower than half the with of the chicken stroke to avoid self intersections and artifacts.



EDIT

Here is an example file. Have in mind that if this is for 3D printing or other Computer aided manufacturing process, there may be geometric issues with production. Shape cleaned up and slightly altered to reduce some of the self intersections. Edit the "Chicken Cookie Cutter" object to control the shape of the section to your liking.